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Recent Advances on Inflation

S. D. Odintsov, V. K. Oikonomou, I. Giannakoudi, F. P. Fronimos, E. C. Lymperiadou

arXiv:2307.16308v1gr-qcastro-ph.CO

TL;DR

The review addresses how inflation can be realized in viable modified-gravity theories while confronting shortcomings of single-field models and new gravitational-wave constraints. It synthesizes scalar, string-inspired, Chern-Simons, f(R), and related models, and discusses how modified gravity changes primordial gravitational-wave waveforms. The review identifies stringent constraints from tensor propagation and the possible blue tilt associated with the NANOGrav signal, while presenting model-dependent inflationary and waveform analyses.

  • Problem

    Single-field inflation has unresolved coupling requirements for reheating, while a cosmological interpretation of the 2023 NANOGrav signal requires a blue tensor tilt that single-scalar-field models cannot explain.

  • Method

    The review synthesizes inflationary dynamics across scalar, string-inspired, Horndeski, Chern-Simons, vacuum f(R), f(R,φ), and kinetic-corrected f(R,φ) theories, including tensor-waveform calculations.

  • Results

    The reviewed frameworks show that viable inflationary predictions and gravitational-wave signals depend on modified-gravity structure, with tensor propagation constraints ruling out some subclasses unless Qf = 0.

  • Takeaways & Limitations

    Future CMB B-mode and gravitational-wave observations may distinguish inflationary theories and constrain the theory underlying a possible cosmological NANOGrav signal.

Abstract

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We review recent trends on inflationary dynamics in the context of viable modified gravity theories. After providing a general overview of the inflationary paradigm emphasizing on what problems of hot Big Bang theory inflation solves, and a somewhat introductory presentation of single field inflationary theories with minimal and non-minimal couplings, we review how inflation can be realized in terms of several string motivated models of inflation, which involve Gauss-Bonnet couplings of the scalar field, higher order derivatives of the scalar field, and some subclasses of viable Horndeski theories. We also present and analyze inflation in the context of Chern-Simons theories of gravity, including various subcases and generalizations of string corrected modified gravities which also contain Chern-Simons correction terms, with the scalar field being identified with the invisible axion, which is the most viable to date dark matter candidate. We also provide a detailed account of vacuum $f(R)$ gravity inflation, and also inflation in $f(R,φ)$ and kinetic-corrected $f(R,φ)$ theories of gravity. In the end of the review we discuss the technique for calculating the overall effect of modified gravity on the waveform of the standard general relativistic gravitational wave form.

I. INTRODUCTION

The review motivates inflation as a response to unresolved hot Big Bang problems and surveys scalar-field and modified-gravity approaches constrained by increasingly precise cosmological and gravitational-wave observations.

  • I. INTRODUCTION: Precision cosmology is expanding through CMB and gravitational-wave observations, including NANOGrav and planned experiments such as CMB-S4, LISA, and the Einstein Telescope.These observations are presented as increasingly important tests of primordial physics.
  • I. INTRODUCTION: Inflation is presented as the leading framework for primordial evolution and large-scale structure because it produces a nearly scale-invariant primordial power spectrum.The review also describes inflation as addressing problems of standard hot Big Bang cosmology.
  • I. INTRODUCTION: Single-field inflation requires couplings between the inflaton and Standard Model particles for reheating, introducing many unexplained parameters and motivating alternatives.The review therefore considers modified-gravity descriptions alongside canonical scalar-field models.
  • I. INTRODUCTION: A cosmological interpretation of the 2023 NANOGrav signal requires a blue tensor tilt, which single-scalar-field models cannot explain and Einstein-Gauss-Bonnet models can achieve only with low reheating temperature.This observation sharply constrains viable inflationary frameworks.
  • I. INTRODUCTION: The review derives slow-roll relations for multiple theories and quantifies modified-gravity effects on tensor perturbations through a single parameter and a multiplicative waveform factor.It aims to provide closed-form relations that allow reproduction of quoted inflationary observables.
  • I. INTRODUCTION: Inflation is described as an early era of abrupt accelerated expansion introduced to resolve shortcomings of the conventional Big Bang model.The review traces this development through early de Sitter and exponential-expansion models.

1. The Horizon problem

The horizon, flatness, and primordial-relic problems arise because standard Big Bang evolution leaves distant regions causally disconnected, requires unstable near-flat initial conditions, and overproduces relics. Inflation addresses these issues through a decreasing comoving Hubble radius and rapid expansion.

  • 1. The Horizon problem: During standard Big Bang evolution, the growing Hubble radius leaves regions observed in the CMB causally disconnected during the primordial era.This conflicts with their observed homogeneity and thermal equilibrium.
  • 1. The Horizon problem: Inflation decreases the comoving Hubble radius, allowing cosmologically relevant regions to have been causally connected before becoming separated.The scale factor grows exponentially while the Hubble rate remains relatively constant.
  • 2. The Flatness Problem: During inflation, the decreasing comoving Hubble radius drives ΩK toward zero, naturally producing an increasingly flat Universe.The review notes that inflation suppresses the density parameter associated with curvature.
  • 3. The Primordial Relics Problem: Primordial relics such as magnetic monopoles would become non-relativistic and dominate over radiation, but observations constrain their present number density to nM(t0) ∼10^-19 cm^-3.Inflation dilutes relics through rapid exponential expansion, making them nearly undetectable.

B. Conditions for Inflation

Inflation requires accelerated expansion, commonly realized through a slowly rolling scalar field whose potential dominates its kinetic energy. The review formulates these conditions using Hubble and potential slow-roll parameters and relates them to observable perturbation quantities.

  • B. Conditions for Inflation: Inflation occurs when the scale factor accelerates, equivalently when the Hubble radius decreases, with near-constant H producing approximately exponential expansion.Exactly constant H gives de Sitter expansion, while time-dependent H gives quasi-de Sitter evolution.
  • B. Conditions for Inflation: The framework begins with a minimally coupled scalar-field action containing the Einstein-Hilbert term, canonical kinetic term, and potential V(ϕ).Variation yields the scalar-field equation and energy-momentum tensor used in the inflationary analysis.
  • B. Conditions for Inflation: In canonical scalar-field inflation, slow roll requires the kinetic term to remain sub-leading to the potential energy.This condition implies a slowly evolving field and |Ḣ| ≪ H^2.
  • B. Conditions for Inflation: The slow-roll indices quantify inflationary evolution, with ε1 ≪1 ensuring inflation and ε2 ≪1 ensuring that it lasts sufficiently long.The end of inflation occurs when the inflationary condition is violated and the perturbative expansion breaks down.
  • B. Conditions for Inflation: For minimal single-field inflation, scalar and tensor observables are expressed through slow-roll parameters, including nS = 1 −4ε1 −2ε2 and r = 16ε1.The inflaton dominates the energy density in the flat FRW setup considered.

1. Massive Scalar Field

The review evaluates massive, self-interacting, and natural single-field inflation models through slow-roll predictions for nS and r. The examples show that observational viability depends strongly on the potential and coupling structure.

  • 1. Massive Scalar Field: For N ≃60, one massive-field example predicts nS ≃0.9889 and r ≃0.044, compared with Planck constraints nS = 0.9649 ± 0.0042 and r < 0.064.The quoted tensor-to-scalar ratio satisfies the stated observational bound, while the spectral-index comparison is presented alongside the Planck constraints.
  • 1. Massive Scalar Field: Power-law potentials in minimally coupled single-field models can be incompatible with observations because their predicted nS and r deviate from the measured ranges.The review notes that non-minimal couplings could make some such models viable under suitable circumstances.
  • 2. Self Interacting Scalar Field: For N ≃60, a self-interacting power-law potential predicts nS ≃0.95 and r ≃0.26 and is judged nonviable against Planck data.Both quoted observables are compared with the Planck values given in the review.
  • 3. Natural Inflation: Natural inflation represents the inflaton as a pseudo-Nambu-Goldstone boson, potentially an axion, with inflation occurring as the field rolls toward the potential minimum.The potential is characterized by scales controlling its height and width.
  • 3. Natural Inflation: Natural inflation requires ϕk ≲0.1MP l to obtain N ≳70 e-folds, and its small-f limit connects to kinetic axion models.The review states that a Taylor expansion for f ≪1 produces the corresponding power-law behavior.
  • Inflationary Observables: The standard observables include scalar and tensor spectral indices and the tensor-to-scalar ratio, with canonical slow roll giving nS = 1 −6ε + 2η, nT = −2ε, and r = 16ε.Modified-gravity models introduce additional slow-roll parameters and nontrivial scalar and tensor propagation speeds.

C. Non-minimally coupled Scalar field Inflation

Non-minimally coupled inflation extends scalar-field models by allowing a field-dependent coupling to curvature. Under slow roll, the review derives analytic observables for general coupling functions and discusses attractor-like behavior.

  • Model formulation: Non-minimal models introduce a scalar-dependent curvature coupling f(R,ϕ)=f(ϕ)R, with f(ϕ) potentially approaching unity at late times.The formulation is presented in the Jordan frame and can be transformed to the Einstein frame.
  • Slow-roll formalism: The additional coupling function introduces slow-roll indices ϵ3 and ϵ4 alongside the conventional parameters ϵ1 and ϵ2.These parameters allow the observables to be expressed under the slow-roll condition ϵi ≪ 1.
  • Observables: For any specified f(ϕ), the slow-roll expressions permit analytic calculation of the tensor-to-scalar ratio r and scalar spectral index nS.The review derives the relevant equations of motion and an approximation for Qs before obtaining r and nS.
  • Example and attractor behavior: Choosing n = 2√3 yields relations for nS and r matching the attractor behavior derived in α-attractor models.The review distinguishes this regime from strongly coupled non-minimal theories, which use different parameter choices and criteria.

IV. A BRIEF ACCOUNT OF THE SWAMPLAND CRITERIA

The review briefly relates inflationary model building to Swampland criteria and then surveys string-inspired corrections, emphasizing their effects on perturbation propagation and tensor spectra. Compatibility with gravitational-wave observations imposes strong restrictions on viable couplings.

  • Swampland criteria: The Swampland discussion introduces distance and de Sitter criteria as separate conditions constraining scalar-field ranges and potential slopes.The review notes that the criteria need not be satisfied simultaneously.
  • Constant-roll evolution: The review states that constant-roll evolution includes the scalar field’s second derivative in the continuity equation, unlike the usual slow-roll treatment.The auxiliary parameter β need not be constant for a ϕ-dependent constant-roll condition.
  • Perturbation propagation: String corrections modify the propagation velocity of scalar and tensor perturbations, requiring well-behaved sound speeds for viable inflationary models.The tensor speed is constrained by 0 < cT ≤ 1.
  • Observational constraints: GW170817 constrains string-corrected models because its gravitational waves propagated at light speed, but imposing Qf = 0 can restore compatibility.This condition reduces the degrees of freedom and constrains the Gauss-Bonnet coupling function.
  • Tensor spectrum: String corrections can produce a blue tensor spectrum when ϵ5 < −ϵ1, whereas the canonical scalar-field model cannot produce such a tensor tilt.The blue spectrum affects the gravitational-wave energy spectrum, especially at high frequencies.

A. Gauss Bonnet Model

The Gauss-Bonnet model uses exponential potential and scalar-coupling functions whose product is constant, allowing the coupling to dominate when the potential becomes small. Its analysis relies on slow evolution of the scalar field and derives inflationary observables from the resulting dynamics.

  • Dilaton model: The dilaton example assigns exponential forms to both the potential and Gauss-Bonnet coupling, with amplitudes V0 and ξ0 controlling their magnitudes.The auxiliary parameter f has mass dimension eV, while V0 has dimension eV4 and ξ0 is dimensionless.
  • Coupling behavior: The product V(ϕ)ξ(ϕ)=V0ξ0 remains constant, while the Gauss-Bonnet coupling dominates in the limit f ≪ 1.These models are motivated as a possible link between early- and late-time cosmological eras.
  • Inflationary dynamics: A slowly varying scalar field is used to obtain a prolonged inflationary era and expressions for the scalar spectral index and tensor-to-scalar ratio.The slow-roll condition is described as sufficient but not mandatory for studying inflation in this framework.

B. Constrained Einstein-Gauss-Bonnet Model

The constrained Einstein-Gauss-Bonnet models impose light-speed tensor propagation through a differential constraint on the coupling function. Specific parameter choices then yield observationally viable inflation, including both blue- and red-tilted tensor spectra.

  • Propagation constraint: Imposing tensor propagation at the speed of light constrains the Gauss-Bonnet scalar coupling function through a differential equation.The constraint reduces the model’s degrees of freedom and motivates the subsequent examples.
  • Potential construction: The exponential Gauss-Bonnet model is combined with power-law behavior because the unconstrained exponential coupling can lead to eternal or absent inflation depending on φ.The resulting potential includes power-law and exponential factors, with further power-law terms appearing during horizon crossing.
  • First constrained example: For φ = 0.01MP, n = 1/2, f0 = 100, and N = 60, the model gives nS = 0.967045, r = 0.000551, and nT = 0.000069.The review reports these values as consistent with current observations and identifies the tensor spectrum as blue tilted.

VII. CHERN-SIMONS AXIONIC GRAVITY

Chern–Simons couplings modify tensor perturbations in polarization-dependent ways, allowing blue tensor spectra and potentially restoring inflationary models otherwise excluded by observations.

  • Tensor perturbations: The gravitational Chern–Simons term alters tensor perturbations through the scalar coupling function and distinguishes left- and right-handed polarization modes.The propagation speed remains equal to the speed of light, consistent with GW170817.
  • Tensor spectrum: A blue tensor spectral index can arise when ϵ5 < −ϵ1, potentially amplifying the primordial gravitational-wave energy spectrum.
  • Chirality: Because the scalar field evolves dynamically, different circular polarizations can acquire different energy-spectrum imprints and enhancement factors.
  • Stability: Chern–Simons models must be checked for ghost instabilities associated with the effective Chern–Simons mass scale and the dynamical scalar coupling.
  • Phenomenological example: For chaotic inflation, suitable Chern–Simons parameters yield nS = 0.96667 and r = 0.00639, consistent with the latest Planck data.
  • Phenomenological example: With a quadratic Chern–Simons coupling, nT ≃4 × 10−8 is positive, while increasing the coupling exponent raises nT and lowers the tensor-to-scalar ratio.The polarization dependence can rectify models previously discarded by observational constraints.

VIII. VACUUM f(R) GRAVITY INFLATION

Vacuum f(R) inflation derives slow-roll observables from modified gravitational dynamics and organizes viability around the parameter x, whose value controls the relation between nS and r.

  • Framework: Vacuum f(R) inflation is formulated through metric field equations, slow-roll indices, and closed expressions for the scalar spectral index and tensor-to-scalar ratio.
  • Parameter evolution: The parameter x evolves with the Hubble rate and its derivatives, and obtaining x(N, σ) generally requires solving the Friedmann equations.
  • Limiting cases: For x ≪1, the leading-order r−nS relation matches the R2 model with fRRR = 0.
  • Viability constraints: Values x ∼4 can make the r−nS relation blow up, while x ≫1 violates the slow-roll condition for ϵ3.
  • Viability constraints: Constant-x models can yield viable slow-roll inflation for x ≪1 and some x ∼4, but their phenomenology remains model dependent.The viable ranges are assessed using Planck constraints and N ∼50−60.

IX. SCALAR FIELD ASSISTED F(R) GRAVITY

Adding a canonical scalar field to f(R) gravity supplies a framework for calculating inflationary dynamics and observables, with viable parameter ranges in representative models.

  • Framework: The scalar-assisted f(R) framework combines a gravitational term, scalar kinetic term, and potential, then derives field equations and slow-roll observables.
  • Inflationary evolution: Initial and final field values are related to model parameters and the e-foldings number through slow-roll dynamics and the condition ϵ1(ϕend) = 1.
  • Phenomenological examples: A power-law potential that is nonviable in usual Einstein–Hilbert gravity becomes viable in the scalar-assisted f(R) model for parameter ranges satisfying Planck 2018 constraints.
  • Phenomenological examples: For V0 = 9.37 × 10−13, β = 6.8 × 10−6, and N = 60, the model gives nS = 0.96611, r = 0.063968, and Pζ(k) = 2.19216 × 10−9.
  • Phenomenological examples: The example imposes an upper bound N = 67 because larger e-folding numbers cannot satisfy the spectral-index, amplitude, and approximation constraints simultaneously.
  • Additional models: For αR gravity, the scalar spectral index is expressed as nS −1 = −4ϵ1 −2ϵ2 = 1 + 2αη −6αϵ, and parameter choices can satisfy observational constraints.

X. INFLATION IN THE CASE OF K-ESSENCE f(R) GRAVITY

This section examines inflation from vacuum f(R) gravity combined with nonquadratic kinetic terms, using slow-roll relations to test observational viability. Some parameter choices satisfy Planck 2018 constraints, but viability can require fine tuning or occur only in effective ghost-containing theories.

  • Model setup: Nonquadratic kinetic terms can produce inflation without a scalar potential when combined with vacuum f(R) gravity.The section studies models where inflation arises from kinetic structure alongside f(R) gravity rather than from a potential.
  • Method: The viability procedure evolves the Hubble rate, slow-roll indices, and observables after specifying f(R) and the kinetic function G(X), then compares nS and r with Planck 2018 constraints.The analysis uses the Friedmann equations and field equation to determine the evolution and observational parameters.
  • Ghost-free case: For (N, n, ti)=(60, 1.36602, 10^-25), one parameter choice gives nS = 0.965 and r = 0.06, satisfying the stated Planck constraints.The example uses (f1, α)=(2.03291, 4.59843×10^-15), within the reported viable ranges.
  • Phantom case: A phantom formulation also yields nS = 0.966 and r = 0.0613, but only with extreme parameter fine tuning and ghost degrees of freedom for f1 > 0.The cited example is compatible with Planck 2018 constraints, while the text characterizes the physical theory as ghost-containing.
  • Reconstruction: In the reconstruction approach, the phantom model admits viable parameter ranges, whereas the ghost-free model cannot satisfy the nS and r constraints simultaneously for the chosen Hubble evolution.The reconstruction expresses the Hubble rate through e-foldings and the Ricci scalar before determining f(R) and the observables.

XI. GENERALIZED f(R, ϕ, X) GRAVITY

This section formulates a generalized f(R, ϕ, X) framework that encompasses canonical, k-essence, string-corrected, and Chern-Simons cases. The unified description supports analysis of noncanonical dynamics, tensor perturbations, and primordial non-Gaussianities.

  • Unified framework: The generalized action depends on the Ricci scalar, scalar field, and kinetic term, with an additional correction Lagrangian for string or axionic contributions.The framework uses X = 1/2 gµν∇µϕ∇νϕ and allows correction terms to enter the dynamics.
  • Special cases: Canonical scalar-field and k-essence models arise as special choices of f(R, ϕ, X), while the Chern-Simons case is obtained by setting the correction stress tensor to zero.These identifications place several inflationary theories within the same background-equation framework.
  • Chern-Simons case: For gravitational Chern-Simons corrections, the background equation of motion remains unaffected while tensor perturbations are modified.The correction contributes differently from string corrections: its principal effect in the cited passage is on tensor perturbations.
  • Noncanonical applications: The framework can describe noncanonical theories such as Dirac-Born-Infeld models and higher-order kinetic terms, including analyses of primordial scalar non-Gaussianities.Under constant roll, the sound-wave velocity can become quite small, affecting the nonlinear phenomenology.
  • Observables and consistency: The sound-wave velocity is required to satisfy 0 < cA ≤ 1, and the generalized formalism expresses spectral indices and tensor-to-scalar ratios through slow-roll quantities.The review presents the auxiliary parameters and slow-roll structure needed to calculate observational predictions.

A. Kinetic Axion f(R) Model

The kinetic axion f(R) model combines axion dynamics with f(R) gravity, producing a primordial stiff-matter phase whose duration depends on reheating. The model can connect primordial behavior to dark-matter-like late-time redshifting, but sufficiently high reheating temperatures spoil observational viability.

  • Observational interpretation: The f(R) sector can preserve viable scalar-spectrum predictions despite ϵ2 ≃ −3, showing that these indices should not automatically be interpreted as ordinary slow-roll parameters.The text attributes the rescue of the model to the f(R) contribution.
  • Reheating and background: The axion stiff-matter phase prolongs inflation depending on the reheating temperature, while the vacuum R2 sector dominates the primordial inflationary background.The reheating-era radiation density is related to temperature through ρreh = π^2 g*T^4reh / 30.
  • Reheating constraint: For TR ∼ O(10^12) GeV, the e-folding number increases to N = 65.3439, with nS = 0.969393 and r = 0.00281042.The reported scalar spectral index lies outside the stated viability region, despite the reduced tensor-to-scalar ratio.

B. Kinetic Axion Gauss-Bonnet Model

The kinetic axion Gauss-Bonnet model incorporates string-corrective Gauss-Bonnet coupling into an f(R) inflationary framework. Under tensor-speed constraints, the scalar field can alter observables, but the model generally cannot produce a blue tensor tilt unless string corrections dominate.

  • Model setup: The model adds a non-minimal scalar-curvature coupling through the Gauss-Bonnet topological density to a kinetic axion assisted f(R) theory.The coupling function is assumed linear, ξ(ϕ) = ϕ, in the considered case.
  • Model setup: The scalar field affects the scalar spectral index through indices ϵ2 and ϵ4, while the R2 term dominates the background equations.The auxiliary parameters x and y are introduced to describe the scalar and Gauss-Bonnet contributions.
  • Tensor constraints: For viable small x and y, the model cannot produce a blue-tilted tensor spectrum unless string corrections dominate over the f(R) contribution.Tensor perturbations are then most likely governed by the f(R) sector, preserving r = −8nT.
  • Observational results: For M = 1.25 × 10^-5MP, N = 60, and f = 10^-5MP, the model gives nS = 0.96686, r = 0.00327847, and nT = −0.000413283.These values are reported as consistent with experimental data and coincide for nT and r with the vacuum R2 result.
  • End of inflation: Imposing the tensor propagation-speed constraint can force the scalar kinetic term to become negligible at inflation’s end, so no extension of inflation occurs.The constraint changes the scalar continuity equation from second order to first order and yields ˙ϕ(tend) = 0.

C. Kinetic Axion Chern-Simons Model

The kinetic axion Chern-Simons model adds a parity-odd gravitational coupling with quadratic scalar coupling. Unlike the preceding cases, it can generate a blue tensor tilt while leaving the scalar spectral index and tensor-to-scalar ratio unchanged.

  • Model setup: The model extends the kinetic axion setup by adding a parity-odd Chern-Simons term with a quadratic coupling function ν(ϕ).The Chern-Simons degree of freedom does not participate in the background equations.
  • Observational results: For f = 10^-9MP and ϕk = 10^-10MP, the tensor spectral index becomes nT = 0.0132771, while the scalar spectral index and tensor-to-scalar ratio remain unchanged from vacuum R2.The blue tilt is enabled by a large ˙ν during first horizon crossing without spoiling f(R) dominance in the background equations.
  • Reheating: Because the Chern-Simons term does not affect the background equations, axion kinetic dominance can produce a stiff-matter reheating era and prolong inflation depending on reheating temperature.This behavior is reported as analogous to the kinetic axion case.

XII. ENERGY SPECTRUM OF PRIMORDIAL GRAVITATIONAL WAVES

The section derives the primordial gravitational-wave energy spectrum and explains how modified gravity can alter its amplitude, frequency dependence, and polarization relative to general relativity. High-frequency modes are especially informative because they retain information about the early Universe, including reheating and inflationary model dependence.

  • Tensor perturbations: The review first formulates tensor perturbations as symmetric, traceless, transverse fields and quantizes their Fourier modes using creation and annihilation operators.These conditions ensure that the perturbations describe gravitational waves, while the commutation relations enable computation of their power spectrum.
  • Energy spectrum: The tensor power spectrum is obtained from vacuum expectation values, after which the gravitational-wave energy density follows from the energy-stress tensor.The resulting general-relativistic spectrum depends on the critical density, the present Hubble rate, transfer functions, and the primordial tensor spectrum.
  • Early-Universe sensitivity: High-frequency gravitational-wave modes probe the early Universe because they re-entered the horizon early, making them sensitive to reheating and inflationary dynamics.The primordial tensor spectrum is model dependent through the tensor spectral index and tensor amplitude, while the exact Hubble-rate value affects the spectrum’s amplitude.
  • Model signatures: Blue-tilted models can amplify the high-frequency spectrum, near-general-relativistic models can produce only small enhancements, and heavy red spectra can suppress primordial signals.Chern-Simons models additionally predict two circular polarizations with different amplitudes at a given frequency.
  • Modified-gravity effects: Modified gravity changes the gravitational-wave spectrum through effects including Planck-mass running, reheating constraints, and modifications to the tensor perturbations.The high-frequency amplitude differs from general relativity when the model-dependent Planck mass is nontrivial, although f(R) gravity may also affect low frequencies.

XIII. CONCLUSIONS

The conclusions present modified gravity as a framework for recent inflationary dynamics and identify forthcoming CMB and gravitational-wave observations as tests of inflation and its possible models.

  • Conclusions: The review surveys modified-gravity descriptions of inflation motivated by shortcomings of single-scalar-field general relativity, including reheating and strongly blue tensor spectra.It also discusses modified gravity as relevant to late-time scenarios with a slightly phantom background equation of state.
  • Scope: The review’s stated contribution is a modern account of inflationary dynamics in viable modified gravity theories.The conclusion frames these theories as extensions of general relativity for describing the inflationary era.
  • Future tests: Future stage-4 CMB, LISA, and Einstein Telescope observations are expected to provide evidence about inflation and clues about the correct inflationary model.The paper identifies direct observation of CMB B-modes as the smoking gun for inflation.
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